High Moments of Large Wigner Random Matrices and Asymptotic Properties of the Spectral Norm

We consider the Wigner ensemble of n×n real symmetric random matrices of the form A (n) ij = 1 √ n aij , whose entries {aij}1≤i≤j≤n are independent random variables with the same symmetric probability distribution such that Eaij = v , and study the corresponding ensemble of random matrices with truncated random variables  (n) ij = 1 √ n â (n) ij , where âij = aijI[−Un,Un](aij). Our main result is that if Un = n 1/6−δ0 with arbitrary fixed 0 < δ0 < 1/6, then the moments M (n) 2k = E { Tr (Â) } obey an asymptotic bound in the limit k, n → ∞ such that k = O(n). This gives the corresponding bound for the probability distribution of the maximal eigenvalue λmax( ) in the limit n → ∞. Restriction supi,j E|aij | 12+2δ0 < ∞ implies the asymptotic equality A =  with probability 1 as n → ∞. In this case the probability distribution of λmax(A ) exhibits the same large deviations bound as the one of the truncated ensemble. The proof is based on the completed and improved version of the method proposed by Ya. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. The bound we obtain is non-universal in the sense that it depends on the probability distribution of aij .